let P be Instruction-Sequence of SCMPDS; :: thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS))
let s be State of SCMPDS; :: thesis: for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS))
let I be Program of SCMPDS; :: thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS))
let a be Int_position ; :: thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS))
let k1 be Integer; :: thesis: ( s . (DataLoc ((s . a),k1)) = 0 implies IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS)) )
set b = DataLoc ((s . a),k1);
assume A1: s . (DataLoc ((s . a),k1)) = 0 ; :: thesis: IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS))
set SAl = Start-At (((card I) + 2),SCMPDS);
set i = (a,k1) <>0_goto 2;
set j = goto ((card I) + 1);
set IF = if<>0 (a,k1,I);
set pIF = stop (if<>0 (a,k1,I));
set s3 = Initialize s;
set P3 = P +* (stop (if<>0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1);
set s5 = Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),2);
set P4 = P +* (stop (if<>0 (a,k1,I)));
set P5 = P +* (stop (if<>0 (a,k1,I)));
A3: if<>0 (a,k1,I) = ((a,k1) <>0_goto 2) ';' ((goto ((card I) + 1)) ';' I) by SCMPDS_4:16;
A4: stop (if<>0 (a,k1,I)) c= P +* (stop (if<>0 (a,k1,I))) by FUNCT_4:25;
then A5: stop (if<>0 (a,k1,I)) c= P +* (stop (if<>0 (a,k1,I))) ;
A6: stop (if<>0 (a,k1,I)) c= P +* (stop (if<>0 (a,k1,I))) by A4;
A7: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
A8: IC (Initialize s) = 0 by MEMSTR_0:47;
A9: Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) <>0_goto 2),(Initialize s)) by A3, Th22 ;
not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= 0 by A1, A7, FUNCT_4:11 ;
then A10: IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1)) = succ (IC (Initialize s)) by A9, SCMPDS_2:55
.= 0 + 1 by A8 ;
A11: 1 in dom (if<>0 (a,k1,I)) by Lm9;
then 1 in dom (stop (if<>0 (a,k1,I))) by COMPOS_1:62;
then A12: (P +* (stop (if<>0 (a,k1,I)))) . 1 = (stop (if<>0 (a,k1,I))) . 1 by A5, GRFUNC_1:2
.= (if<>0 (a,k1,I)) . 1 by A11, COMPOS_1:63
.= goto ((card I) + 1) by Lm10 ;
A13: (P +* (stop (if<>0 (a,k1,I)))) /. (IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1))) = (P +* (stop (if<>0 (a,k1,I)))) . (IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1))) by PBOOLE:143;
A15: Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),(1 + 1)) = Following ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1))) by EXTPRO_1:3
.= Exec ((goto ((card I) + 1)),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1))) by A10, A12, A13 ;
then A16: IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),2)) = ICplusConst ((Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1)),((card I) + 1)) by SCMPDS_2:54
.= ((card I) + 1) + 1 by A10, Th23
.= (card I) + (1 + 1) ;
A17: (P +* (stop (if<>0 (a,k1,I)))) /. (IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),2))) = (P +* (stop (if<>0 (a,k1,I)))) . (IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),2))) by PBOOLE:143;
A19: card (if<>0 (a,k1,I)) = (card I) + 2 by Lm8;
then (card I) + 2 in dom (stop (if<>0 (a,k1,I))) by COMPOS_1:64;
then (P +* (stop (if<>0 (a,k1,I)))) . ((card I) + 2) = (stop (if<>0 (a,k1,I))) . ((card I) + 2) by A6, GRFUNC_1:2
.= halt SCMPDS by A19, COMPOS_1:64 ;
then A20: CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),2))) = halt SCMPDS by A16, A17;
then A21: P +* (stop (if<>0 (a,k1,I))) halts_on Initialize s by EXTPRO_1:29;
A22: CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s)) = (a,k1) <>0_goto 2 by A3, Th22;

now

let l be Element of NAT ; :: thesis: ( l < 1 + 1 implies CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),b₁))) <> halt SCMPDS )
assume l < 1 + 1 ; :: thesis: CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),b₁))) <> halt SCMPDS
then A23: l <= 1 by NAT_1:13;
A24: Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),0) = Initialize s by EXTPRO_1:2;
A25: (P +* (stop (if<>0 (a,k1,I)))) /. (IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),l))) = (P +* (stop (if<>0 (a,k1,I)))) . (IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),l))) by PBOOLE:143;

per cases ( l = 0 or l = 1 ) by A23, NAT_1:25;

suppose l = 0 ; :: thesis: CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),b₁))) <> halt SCMPDS

then CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),l))) = CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s)) by A24;
hence CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS by A22; :: thesis: verum

end;

suppose l = 1 ; :: thesis: CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),b₁))) <> halt SCMPDS

hence CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS by A10, A12, A25; :: thesis: verum

end;

end;

end;

then for l being Element of NAT st CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),l))) = halt SCMPDS holds
2 <= l ;
then LifeSpan ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s)) = 2 by A20, A21, EXTPRO_1:def 15;
then A27: Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),2) = Result ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s)) by A21, EXTPRO_1:23;

A29: now

A30: dom (Start-At (((card I) + 2),SCMPDS)) = {(IC )} by FUNCOP_1:13;
let x be set ; :: thesis: ( x in dom (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) implies (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b₁ = (s +* (Start-At (((card I) + 2),SCMPDS))) . b₁ )
assume A31: x in dom (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) ; :: thesis: (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b₁ = (s +* (Start-At (((card I) + 2),SCMPDS))) . b₁

per cases ( x is Int_position or x = IC ) by A31, SCMPDS_4:6;

suppose A32: x is Int_position ; :: thesis: (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b₁ = (s +* (Start-At (((card I) + 2),SCMPDS))) . b₁

then x <> IC by SCMPDS_2:43;
then A33: not x in dom (Start-At (((card I) + 2),SCMPDS)) by A30, TARSKI:def 1;
TT: not x in dom (Start-At (0,SCMPDS)) by A32, SCMPDS_4:18;
thus (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . x = (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),2)) . x by A27
.= (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1)) . x by A15, A32, SCMPDS_2:54
.= (Initialize s) . x by A9, A32, SCMPDS_2:55
.= s . x by TT, FUNCT_4:11
.= (s +* (Start-At (((card I) + 2),SCMPDS))) . x by A33, FUNCT_4:11 ; :: thesis: verum

end;

suppose A34: x = IC ; :: thesis: (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b₁ = (s +* (Start-At (((card I) + 2),SCMPDS))) . b₁

hence (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . x = (card I) + 2 by A16, A27
.= (s +* (Start-At (((card I) + 2),SCMPDS))) . x by A34, FUNCT_4:113 ;
:: thesis: verum

end;

end;

end;

dom (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) = the carrier of SCMPDS by PARTFUN1:def 2
.= dom (s +* (Start-At (((card I) + 2),SCMPDS))) by PARTFUN1:def 2 ;
hence IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS)) by A29, FUNCT_1:2; :: thesis: verum